evaluate the limit, n->infty, ( sum[i=1,n] { f(ci) delta-xi } )

[nbg273]

Use Example 1 as a model to evaluate the limit


\(\displaystyle \lim _{n\to \infty }\left(\sum _{i=1}^n\left(f\left(ci\right)Δxi\right)\right)\)


over the region bounded by the graphs of the equations. (Round your answer to three decimal places.)


f(x) = sqrt(x), y = 0, x = 0, x = 3


HINT: Let \(\displaystyle ci=\frac{3i^2}{n^2}\)


Can someone lead me in the right direction?

 

[HallsofIvy]

You write "Use Example 1 as a model to evaluate the limit" but there is no "Example 1" in what you write!

Can you write the sum for general n? With \(\displaystyle c_i= \frac{3i^2}{n^2}\), \(\displaystyle f(c_i)= \frac{9i^4}{n^4}\) and \(\displaystyle \Delta x_i= c_i- c_{i-1}= \frac{3i^2}{n^2}- \frac{3(i-1)^2}{n^2}= \frac{3(2i- 1)}{n^2}\) so that \(\displaystyle f(c_i)\Delta x_i= \frac{9i^4}{n^4}\frac{3(2i- 1)}n^2= 27\frac{i^4(2i- 1)}{n^2}\). You will need to know some formulas for powers of i to evaluate that. Generally, the sum of the kth power of i, for i from 1 to n, will be a k+ 1 polynomial in n.